Nam Nguyen wrote:> On 08/05/2013 8:11 AM, Nam Nguyen wrote:>> On 08/05/2013 7:28 AM, Frederick Williams wrote:>>> Nam Nguyen wrote:>>>>>>>> On 05/05/2013 8:45 AM, Frederick Williams wrote:>>>>> Nam Nguyen wrote:>>>>>>>>>>>> On 04/05/2013 10:07 AM, Frederick Williams wrote:>>>>>>> Nam Nguyen wrote:>>>>>>>>>>>>>>>> On 26/04/2013 11:09 AM, Nam Nguyen wrote:>>>>>>>>>>>>>>>> On 2013-04-25, FredJeffries <fredjeffries@gmail.com> wrote:>>>>>>>>>>>>>>>>>>>> Now PA has been proved consistent in ZF or NBG, but then that>>>>>>>>>> brings the consistency of axioms for set theory.>>>>>>>>>>>>>>>> Exactly right. And exactly my point.>>>>>>>>>>>>>>>> Somewhere, somehow, a circularity or an infinite regression>>>>>>>> of _mathematical knowledge_ will be reached,>>>>>>>>>>>>>> How does one reach an infinite regression?>>>>>>>>>>>> By claiming that the state of consistency of PA can be>>>>>> proved _IN_ a _different formal system_ .>>>>>>>>>> Your notion of infinite is very modest if does not go beyond two.>>>>>>>> That does _not_ mean there be only two, actually.>>>>>>>>>>>>>>>>>>>> and at that point>>>>>>>> we still have to confront with the issue of mathematical>>>>>>>> relativity.>>>>>>>>>>>>>> It is not the case that either we go round in a circle or we regress>>>>>>> forever.>>>>>>>>>>>> That's not a refute. Of course.>>>>>>>>>>>> (It's just an unsubstantiated claim).>>>>>>>>>> And yet an obviously true one. Suppose the question of the>>>>> consistency>>>>> of PA is raised, a party to the discussion may say 'I accept that>>>>> PA is>>>>> consistent and I feel no need to prove it.' No circle, no regression.>>>>>>>> The circularity rests with the argument on the _actual and objective_>>>> state of consistency of PA, _not_ on the _wishful and subjective_>>>> "acceptance" of anything.>>>>>> Mathematicians (like the rest of humanity) are forever accepting>>> things. It is no big deal.>>>>> Verification, proving, is a big deal.>> For example, would you _accept_ the consistency of PA + ~cGC> ("It is no big deal" you said)?

Not everyone shares your obsessions.

The consistency of PA may be an objective fact (or fiction), but proving is a human activity.